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INVITED LECTURES

Charles R. Johnson

"Special classes of matrices: positivity"

Primary Classes: positive definite matrices (PD), M-matrices (M), totally positive matrices (TP), and P-matrices (P)

Purpose: to acquaint researchers with the fundamental properties of and recent results about these classes of matrices to support research in this major area of matrix analysis. Open problems will be mentioned.

Necessary Background: elementary linear algebra, basic theory of eigenvalues, and some familiarity with the Perron-Frobenius theory of nonnegative matrices

Lecture 1: Introduction to each class, including reasons for interest, basic properties and some open questions. The major tool for each class (Schur parameters for PD, diagonal dominance for M, the elementary bidiagonal factorization for TP, and non- sign- reversal for P) will be introduced. Also the related classes, such as doubly nonnegative (DN), inverse-M (IM), positive P, and stable matrices, etc, will be introduced.

Lecture 2: Determinantal inequalities known in each class, their implications, and further questions about determinantal inequalities. The unifying role of Newton matrices and the Taussky unification problem.

Lecture 3: Completion and interpolation problems for each class, and open questions in this area.

Lecture 4: Advanced properties for each class, particular recent results, and open questions stemming from recent work.

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